The class blog for Math 3010, fall 2014, at the University of Utah

Googols

During class, we spoke briefly about Graham’s number. It’s a number so vast that it can’t be written in our conventional decimal number system. There are other very large numbers as well, such as the googolplex. We run into a problem when talking about numbers on this scale, because we rarely encounter really large numbers in day to day life. In fact, very few people have a frame of reference for how large numbers like a trillion really are. One of my favorite numbers is the googol. It’s the number that nerdy kids throw around when they want to sound smart when describing innumerable quantities. The definition of a googol is pretty simple, 10100. The one question that remains: What would it take to have a googol?

An easy way to relate large numbers is to compare them with money. Consider a $100 bill. It’s a rather light, thin piece of paper, but it holds real value to a lot of people. Its dimensions are 6.14 in. x 2.61 in. [1], and it is .0043 in. thick. [2]

Assembling a million dollars in one location using $100 bills is easy (if you have the money, of course). If you were to stack them neatly into a cube, the cube would be about a quarter of a foot long, a quarter of a foot wide, and a quarter of a foot high. This is about the size of a basketball.

This may be all well and good, but what about a billion? Each side of the cube now is 7.36 feet long. This cube would also weigh 11 tons. This is about the weight of 3.5 Subaru Foresters.

How about the glorious trillion? A trillion dollars would form a cube 73.6 feet long. If stacked on a football field, the pile of money would be 7 feet deep. The current national debt of the United States is around $17 trillion. This money on a football field would be 117 feet deep. It’s about the money required to half-fill Rice-Eccles stadium with $100 bills.

As the numbers grow, the money pile gets more and more ridiculous. Let’s consider how the short scale number system works. A million is a one with six zeroes. A billion is a one with nine zeroes. Every ‘–illion’ thereafter has an additional three zeroes. How do we determine what should prefix ‘-illion’? We count in Latin. Therefore, we get prefixes like ‘quad-‘, ‘quint-‘, and ‘sept-‘, just to name a few. A googol in the short scale system is 10 duotrigintillion, or ’32-illion’.

A quintillion dollars (that’s $1,000,000,000,000,000,000) would form a cube 7,360 feet long, or about 1.4 miles on each side. This is more money than the purchasing power of the Earth by a factor of 11,300. [3] In addition, if we divided up this money equally among the world’s population, each person gets $140,350,877. This would also likely cause the world’s economy to completely fall apart.

A septillion dollars would form a cube measuring 139x139x139 miles. This is rapidly getting harder to comprehend. This cube of money would weigh 11 quadrillion tons. It’s also enough to completely cover the Earth in money 83 feet deep.

A nonillion dollars would form a cube 13,940 miles across. This is an enormous amount of money, and would in fact form a cube much larger than the Earth.

An undecillion dollars would create a cube 1.4 million miles across, which is larger than the sun. Oddly enough, the bakery chain, Au Bon Pain, was once sued for $2 undecillion dollars. [4] While the lawsuit may have been filed, it was under spurious claims, and was subsequently dropped.

We’re starting to run out of comparable objects. The sun is the largest object that people are most people are familiar with. However, there are still some valid comparisons to be made. 1 AU (Astronomical Unit) is 92,955,806 miles across.[5] It’s defined as the median distance from the Earth to the Sun. One tredecillion dollars (for reference, that’s $1,000,000,000,000,000,000,000,000,000,000,000,000,000,000) would form a cube of $100 bills that is 1.5 AU across. You could line up one corner of the cube on the center of the sun and have the other corner reach Mars.

By now, we’ve reached Mars from the Sun. If we increase the value of our cube to $1 quindecillion, our cube has grown to the point that we could set a corner of it on the Sun, and have the other corner of the cube reach the Voyager 1 probe, the farthest man-made object from Earth. We’re now at 150 AU. How much further do we need to go to reach a googol?

Well, to get a cube that reaches from the Sun to the center of the Milky Way galaxy, we need $1.2 duovigintillion (That’s 22-illion). This is around 1.7 billion AU. Having exhausted our units again, we’ll need to switch to light-years. There are 63,241 AU in one light-year. One trevigintillion dollars would form a cube 237,000 ly across.

It takes roughly $1.1 quattorvigintillion to form a cube of money that stretches from our sun to the Andromeda Galaxy. We’ve also long passed the point at which the cube of money would collapse to form a black hole. The cube would now be 2,400,000 ly across.

We’re going to need a lot of these. Image by Bureau of Engraving and Printing, via Wikimedia Commons.

Once we reach $4 octovigintillion, we have a cube of money that is the size of the observable universe. Unfortunately, we’ve really run out of reference points. The only thing that we can do now to reach a googol is to increase the value of our bills!

Finally, we’d need to increase the value of our bills to quintillion dollar notes, but at last, we are successful. We filled the observable universe with bills that are worth thousands of times more than the world economy, but we’ve finally gotten a googol! Congratulations!